For a category and object , let denote the full subcategory of whose objects are the monic morphisms. A subobject of is an isomorphism class of objects in .
Theorem: Let and be monics in . and are isomorphic in if and only if and have the same image.
Proof: Suppose and have the same image. Denote . Then in fact and are bijections, considered as and . Take and . Clearly is a morphism in : and similarly for . It’s tirival then that , so and are isomorphic.
Suppose and are isomorphic. Then, we have and that are morphisms in (meaning and ), and furthermore and are (mutually inverse) isomorphisms. The latter fact is actually not needed for this direction. Observe that that the first equality guarantees that anything in the image of is in the image of , and the second guarantees that anything in the image of in the in the image of . So, .
So, we have that subobjects of a set are in canonical correspondence with the subsets of .
In , the situation calls for a bit more subtlety. There exist monics in that that have the same subspace as an image, but are not isomorphic. Consider the space and . In both cases, consider the map . So we have and that are monics with the same image (), but and are not homeomorphic, and hence these are two different subobjects. However, isomorphism classes of embeddings (a continuous injective function that is homeomorphic onto its image) do correspond one-to-one with subspaces.